Almost-prime values of polynomials at prime arguments

Abstract

We consider almost-primes of the form $f(p)$ where $f$ is an irreducible polynomial over $\mathbb{Z}$ and $p$ runs over primes. We improve a result of Richert for polynomials of degree at least $3$. In particular we show that, when the degree is large, there are infinitely many primes $p$ for which $f(p)$ has at most $°f+O(\log °f)$ prime factors.

17 pages

References in corpus (1)

• New equidistribution estimates of Zhang type

Cited by in corpus (4)

• Quadratic polynomials at prime arguments
• Almost-prime values of reducible polynomials at prime arguments
• Almost-Prime Polynomials with Prime Arguments
• On a Diophantine inequality involving a prime and an almost-prime